Name:__________________________
Statistics Unit:
#
1.
Assignment
Mean / Median / Mode pg 2
2.
Assignment #1 pg 5
3.
Weighted and Trimmed Means pg 10
4.
Assignment #2 pg 13
5.
Percentiles pg 17
6.
Assignment #3 20
Completed?
Comments
Statistics Test:_________________________
Page 1 of 23
Lesson 1:
Mean / Median / Mode
Statistics are used in everyday life. They are used to predict weather, figure out
insurance premiums, and show the abilities of athletes. All data that you encounter
is based on statistics, however, this data may have skewed or half true conclusions
depending on what kind of information the user wants to pass on.
You will be dealing with measures of central tendency in this unit, or measurements
based on the center of a group of data.
The word ‘average’ is often used in everyday language to describe the sum of a set
of values divided by the total number of values. In statistics, this term is known as
the arithmetic mean. There are two other common statistical terms that are used
to refer to the center of a set of values – the median and the mode.
The median of a set of values is the middle value when the values are arranged in
ascending or descending order. The mode of a set of values is the value that
occurs the most often.
The range of the data tells you the upper and lower parameters of the given data.
Therefore you would take the highest score and subtract the lowest score to find
out the range of any piece of data.
_
NOTE: The statistical notation for the mean is X .
Example 1:
Calculate the mean, the median, the mode, and the range for the following set of
values:
55, 62, 70, 77, 78, 78
Solution:
Mean =
Median =
Note: if there is an even number of values, you
must average the two values in the middle.
Page 2 of 23
Mode =
(the number that occurs most often)
Range =
(highest number – lowest number)
Example 2:
Ross has a mean mark of 50% for her first three math tests and then he earns a
mark of 70% on his fourth test. Ross states that since the average of 50 and 70
is 60, her new mean math mark is 60%. Do you think Ross is correct?
Solution:
Example 3: Graphing
Graph the following chart on 2 separate graphs, one for Mechanics and one for
Auto body technicians.
Mechanics
Person
1
2
Wage ($) 25 27
3
4
5
6
7
8
9
10
11
12
13
14
27
24 25
24
25
25 35
20
26
26
17
27
7
27
8
28
10
24
11
17
12
35
13
19
14
18
Auto body Techs
Person
1
2
3
Wage($) 23 26 28
4
26
5
17
6
18
9
23
Page 3 of 23
1. Is it easier to estimate the mean of one group over the other? Estimate the
means.
2. Which group has the higher mean wage?
3. What is the range of each set of data?
4. What is the actual mean for each group? Compare?
Page 4 of 23
Assignment #1
1. Over a 5 week period, attendance at a dance lesson per week was 14, 16, 12, 18,
and 12. Calculate:
a. the mean attendance
b. the median attendance
c. the mode
d. the range
2. The table below shows the high and low temperatures recorded for Brandon,
Manitoba in October.
Day
High
Low
MON
12
3
TUE
8
-2
WED
10
-3
THUR
9
-6
FRI
8
-3
SAT
10
3
SUN
9
2
Graph the daily high and the daily low on one graph. Use the day of the week as
your x – axis and the temperature as your y – axis. Estimate the mean and median
high and low temperatures for this week.
Page 5 of 23
Now Calculate:
a. the mean high and low temperature
b. the median high and low temperature
c. the mode
d. the range
Which measurement best helps you decide what type of jacket you are going to
wear that day? Why?
3. Daggie is looking for jobs as a construction worker and found the following
hourly wages in Manitoba:
22.45, 29.50, 11.40, 22.45, 28.75, 19.50, 17.45, 22.20,
29.50, 22.45, 38.50, 16.65, 21.25, 22.20, 17.45, 21.15
a. Calculate the mean, median, mode, and range for the pay rates.
b. Which measure is the better indicator of the salary that Daggie might expect
and why?
Page 6 of 23
4. Eunice is a dressmaker. She spent a mean of $15.87 / meter of fabric. She
spent a total of $5554.50. How many meters of fabric did she buy?
5. Trevor owns a small garbage removal company. Last week, he made 12 visits to
the dump to empty his truck. He charged his customers the following amounts.
$450, $250, $375, $500, $125, $275, $75, $150, $375, $475, $200, $450
a. What is the mean amount Trevor charged his customers / load?
b. What was the median?
c. What was the mode? Explain whether or not this is a good measure of
central tendency?
d. Why would he have charged the customers different amounts?
e. If it cost him $75 to dump each load, how much did he make on the 12 loads?
What were his mean earnings per load?
Page 7 of 23
6. Mark works part-time as a salesman at a car dealership. The following chart
shows his hours that he worked for one month.
Week
Monday Tuesday
Wed
Thurs
Fri
Sat
Sun
1
4.25
4
6
6.5
7.25
2
6
4
8
3
4
6.25
7
8
4
4.25
5
4
4
7.5
a. What was the mean number of hours Mark worked per week over the 4
weeks?
b. Mark added all the hours together and divided by 17 and got an answer of
5.6 hours. What did Mark calculate?
7. Anne works as a server at a local restaurant. Last night she received the
following tips:
$8.55, $12.75, $5.90, $12.60, $3.50, $8, $3.20, $10.50, $8, $3.60, $10,
$15, $8.50
a. What was her mean tip per order?
b. What was her median tip per order?
Page 8 of 23
c. Anne must share her tips with the kitchen staff (25%). After sharing, what
was her mean tip per order?
d. If, during the evening, she served the above orders plus two tables that did
not leave a tip, what was her mean tip per table before and after sharing
with the kitchen staff.
8. Give a good example of when the mean vs the mode vs the median should be
used as a measurement of central tendency.
Page 9 of 23
Lesson #2
Outliers, Weighted and Trimmed Means
Outliers:
An outlier is an observation that is numerically distant from the rest of the data.
It’s a value that lies outside (and is much larger or smaller than) the other values in
a set of data.
Example 1:
There are five people in a group that are 61, 61, 63, 64, 66, and 90 inches tall.
a) Determine the mean, median, and mode.
b) What is the outlier? Remove it from the data and recalculate the mean,
median, and mode.
Trimmed Mean:
The trimmed mean is calculated by discarding a certain number or percentage of
the lowest and highest scores, and then calculating the mean based on the
remaining scores. A trimmed mean is less susceptible to the effects of extreme
scores (outliers) than the arithmetic mean.
Page 10 of 23
Example #2:
Suppose a gymnast in the London Olympics received the following scores.
7.5, 8, 9.5, 6.5, 7, 7.5, 8,
7.5, 8, 7
Her final score is based on removing the highest and lowest scores and then
finding the mean. Calculate the following:
a.
The arithmetic Mean.
b. The trimmed Mean.
Example #3
The trimmed mean can also be found by dropping a percentage of the highest and
lowest scores. Calculate the trimmed mean for Example 2 except this time solve
trimming 10% of the data. (Round final answer to nearest integer)
Step 1: Count the number of observations, labeled n.
Step 2: Reorder the scores from smallest to largest.
Step 3: Find the proportion trimmed from the data.
p = P / 100 where P = % trimmed
Therefore
Step 4: Calculate the total number of scores to be removed.
nXp =
Therefore, we will remove the highest and the lowest
score to find out the mean trimmed 10%.
Page 11 of 23
NOTE: If there were 20 scores then n = 20, p = 20 / 100 = .2
and n X p = .2 X 20 = 4. Therefore you would remove 4 scores
from the top and 4 scores from the bottom.
Weighted Mean:
The weighted mean is often used when giving grades in school. It is similar to the
arithmetic mean except some data points may be more valued than others.
Example #4:
Mr. Krahn has 2 math classes. One class has 5 students while the other has 10.
Each class scored the following grades:
Class 1: 55, 69, 80, 84, 62
Class 2: 70, 90, 55, 84, 88, 93, 78, 69, 98, 75
The mean for class 1 is 70. The mean for class 2 is 80. If you calculate the mean
for both classes you get a mean of 75.
This answer does not account for the different number of students in each class.
The proper mean for both classes can be found by totaling all of the grades and
dividing by 15 students.
Therefore:
OR: You can use the weighted mean:
Mean =
Page 12 of 23
Assignment #2
1. What is the difference between the arithmetic mean and the trimmed mean?
2. Louise makes purses for extra income. The following numbers show how many
hours she spent making each purse.
6, 8, 37, 3, 9, 6, 8.5, 5, 7.5, and 9
a. What is the mean number of hours Louise spent on each purse?
b. What is the trimmed mean of hours spent on each purse if you remove the
highest and lowest number?
c. Which mean best represents the number of hours that Louise spent on the
purses and why?
d. What might explain the outliers in this case?
Page 13 of 23
3. A teacher has two grade 12 math classes. The 25 students in class A scored
the following test results:
65, 75, 92, 53, 87, 59, 32, 80, 76, 37, 68, 79, 67, 69, 81, 57, 66, 71, 90, 73,
90, 72, 61, 67, 53,
The 20 students in Class B scored the following test results:
98, 79, 83, 58, 69, 84, 77, 86, 89, 63, 78, 76, 59, 89, 74, 55, 69, 64, 87, 98
a. What is the arithmetic mean for class A? and B?
b. Calculate the weighted mean for the 2 classes combined using 2 different
methods.
4. Frank is in the Engineering Faculty at U of M and earned the following marks.






8 / 10 on an assignment
7 / 10 on a quiz
7.5 / 10 on a presentation
9 / 10 on the second assignment
10 / 10 on the second quiz
82% on his final exam
Page 14 of 23
His final grade is calculated by the following: 10% for assignments, 15% for
quizzes, 25% for presentations, and 50% for final exams.
What is Frank’s final grade?
5. Bill wants to know what his average score in golf was over a season but is not
sure which mean to use. He golfed at 2 different golf courses during the
summer, here are his scores.
Bridges Golf Course – 75, 82, 83, 77, 68, 88, 86, 84, 79, 80, 84, 95
John Blumberg Course – 77, 76, 74, 67, 75, 74, 99, 80
a. Find his overall mean.
b. Find his mean for each course.
c. Find his weighted mean based on the 2 courses.
Page 15 of 23
d. Find his trimmed mean for both courses if he took his lowest and highest 2
scores away.
e. Find his 20% trimmed mean for both courses.
f. What does all this information say about Bill’s golf game and the courses
that he plays at?
Page 16 of 23
Lesson #3
Percentiles
A Percentile indicates the position of a term in a set of data allowing us to compare
it to all of the others.
For example:
In a college entrance examination a score of 75% might seem quite good. However,
if out of 100 students, 75% was only better than 25 students, that might affect
your chances of getting in. Or, if 75% was better than 80 students, that might also
affect your chances.
25 students
|
75 students
75%
80 students
|
20 students
75%
One way of comparing scores is to use Percentile Rank, or the percentage of
scores that fall below a particular score. Thus, in the above example the first
percentile is only the 25th while the second is the 75th. The higher the percentile
rank the better! It can be calculated as follows:
Percentile Rank = B + 0.5E x 100
N
Where B = the number of scores Below a given score.
Where E = the number of raw scores Equal to the given score, including the given
score. However, if there are n other scores, equal to the given score, then E = 0.
Where n = the total number of raw scores
Page 17 of 23
For Example:
800 applicants wrote a university entrance exam. One applicant scored 65%.
620 applicants scored lower than he did. What is his percentile rank?
Solution:
B=
E=
n=
Percentile Rank =
+
x 100 =
Therefore, _________% of all applicants scored lower than this applicant, or
this applicant scored better than _________% of all applicants. (Always round
percentiles up to the next whole number.)
For Example:
500 job applicants wrote an exam. One applicant scored 65%. 380 applicants
scored lower than she did, and 23 other applicants also scored 65%. What is her
percentile rank?
Solution:
B=
E=
n=
Percentile Rank =
+
X 100 =
Therefore, ______% of all applicants scored lower than this applicant, or this
applicant scored better than _________% of all applicants. (Always round
percentiles up to the next whole number.)
Page 18 of 23
The following notation is used to represent percentile rankP79 is equivalent to the 79th percentile.
P25, P50, and P75 are special percentile ranks that represent the lower
quartile, the median, and the upper quartile, respectively.
25th
Lower quartile
50
Median
75th
Upper Quartile
Percentiles are used quite frequently to describe the results of achievement tests
and the ranking of people taking those tests, including those applying for
government jobs or entrance into certain faculties at universities and community
colleges. If there are more applicants than available positions, candidates are
often ranked according to percentiles. Since percentiles are used quite often,
special names are given to the 25th and 75th percentiles of a distribution. Of
course, the 50th percentile is the median.
The various percentiles are denoted by the letter P with the appropriate subscript.
For example, P20 represents the 20th percentile and P88 the 88th percentile.
Page 19 of 23
Assignment 1: Percentiles
1.
A group of Senior I students was given a differential aptitude test (DAT) in
order to assess their strengths and weaknesses in English and mathematics.
The test consisted of eight sub-tests. Percentile rankings were determined
for each student for each sub-test.
Use the percentile rankings for Student A and Student B to answer the
questions that follow:
a) What aptitude appears to be the strongest for Student A?
b) What percentage of students who wrote this test scored lower than
Student A in Mathematical Skills?
c) For the sub-test in Reading Comprehension, how do Student A and
Student B compare with the rest of the students who wrote this test?
d) With reference to this particular aptitude test, predict the area of
study in which you think Student B would most "likely" be successful.
e) Which aptitude appears to be the weakest for Student A?
Page 20 of 23
2. A total of 3286 students wrote a university entrance examination. Student X
and 432 other students had a score of 891 out of 1200. There were 2279
students who scored lower than 891.
In order to gain entrance, a given student needed a percentile ranking of 70 or
better.
a) What was the percentile ranking of Student X?
b) Did Student X score high enough to gain entrance into the university?
3. In Leanne's band class of 50 students, 26 students can play fewer instruments
than Leanne can, and 4 students can play as many instruments as she can.
a) Find her percentile rank. What does this percentile mean?
b) What percent of students play more instruments than Leanne?
Page 21 of 23
4. The following statistics are available on the family income for the community of
Manwintoba that has a total of 2200 families:
P25 = $15,500
P75 = $42,750
P50 = $28,475
P85 = $64,250
Approximately what percent and how many families earn:
a) less than $28,475?
b) more than $64,250?
c) less than $42,750?
d) more than $15,500?
e) between $15,500 and $64,250?
f) What is the median family income for this community?
5. The following is a set of 40 scores achieved by students on an examination:
Page 22 of 23
Determine the percentile rankings for each of the following scores. (Note: Round
all rankings up to the next whole number.)
a) 41
b) 93
c) 29
e) 65
6. Roberto has a final Senior 4 average of 88%. The college that he wishes to
attend will not accept any applicant if his or her percentile rank is below 85. Can
Roberto be sure that he will be accepted by this college or is it possible that he
will be denied admission? Explain your answer.
7. A student scored 38% on a recent math test. However, the student's
percentile rank on the test was 82. What can you conclude about the success rate
of most other students? What could cause test results like this?
Page 23 of 23